Question

1. find all the unknown angle measures in the diagram. justify your answer for each angle measure.

Explanation:

Step1: Find $\angle2$

Since $\angle2 + 75^{\circ}=90^{\circ}$ (complementary angles), then $\angle2=90^{\circ}-75^{\circ}$.
$\angle2 = 15^{\circ}$

Step2: Find $\angle1$

Since $\angle1+75^{\circ}=90^{\circ}$ (complementary angles), then $\angle1 = 90^{\circ}-75^{\circ}$.
$\angle1=15^{\circ}$

Step3: Find $\angle4$

Since $\angle4 + 30^{\circ}=\angle2 + 75^{\circ}$ (vertical angles), and $\angle2 = 15^{\circ}$, then $\angle4+30^{\circ}=15^{\circ}+75^{\circ}$.
$\angle4+30^{\circ}=90^{\circ}$, so $\angle4=90^{\circ}-30^{\circ}$.
$\angle4 = 60^{\circ}$

Step4: Find $\angle5$

Since $\angle4+\angle5 + 30^{\circ}=90^{\circ}$ (complementary angles), and $\angle4 = 60^{\circ}$, then $60^{\circ}+\angle5+30^{\circ}=90^{\circ}$.
$\angle5=90^{\circ}-60^{\circ}-30^{\circ}$.
$\angle5 = 0^{\circ}$ (This is wrong, actually, since $\angle1 = 15^{\circ}$, and $\angle1+\angle5+30^{\circ}=90^{\circ}$ (complementary angles related to the right - angle formed by the vertical and horizontal lines). So $15^{\circ}+\angle5+30^{\circ}=90^{\circ}$, then $\angle5=90^{\circ}-15^{\circ}-30^{\circ}$.
$\angle5 = 45^{\circ}$

Answer:

$\angle1 = 15^{\circ}$, $\angle2=15^{\circ}$, $\angle4 = 60^{\circ}$, $\angle5 = 45^{\circ}$